Lanjutan Sistem Persamaan Linear Tiga Variabel (Matematika Wajib Kelas X)

 $\text{2. Metode determinan Matriks}$

Perhatikan kemabil bentuk SPLDV dan SPLTV berikut:

$\{\begin{array}{l}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$ 

dan

$\{\begin{array}{l}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{1}x+b_{1}y+c_{1}z=d_1\\ a_{1}x+b_{1}y+c_{1}z=d_1\end{array}$

Metode determinat matriks adalah penyelesaian nilai tidap variabel dengan menggunakan determinan berikut:

Misalkan saja diberikan:

$\begin{array}{rl}& \begin{array}{rl}ax+by& =p\\ cx+dy& =q\end{array}\\ \\ & \text{dan}\\ \\ & \begin{array}{rl}ax+by+cz& =r\\ dx+ey+fz& =s\\ gx+hy+iz& =t\end{array}\end{array}$

maka penyelesaian dengan model matriks adalah:

$\begin{array}{|ccc|}\hline \text{Metode}& \mathbf{\text{SPLDV}}& \mathbf{\text{SPLTV}}\\ \text{Determinan}& \begin{array}{rl}x& ={\displaystyle \frac{\left|\begin{array}{cc}p& b\\ q& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}}\\ & \text{dan}\\ y& ={\displaystyle \frac{\left|\begin{array}{cc}a& p\\ c& q\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}}\\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \end{array}& \begin{array}{rl}x& ={\displaystyle \frac{\left|\begin{array}{ccc}r& b& c\\ s& e& f\\ t& h& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}}\\ & \text{dan}\\ y& ={\displaystyle \frac{\left|\begin{array}{ccc}a& r& c\\ d& s& f\\ g& t& i\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}}\\ & \text{serta}\\ z& ={\displaystyle \frac{\left|\begin{array}{ccc}a& b& r\\ d& e& s\\ g& h& t\end{array}\right|}{\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|}}\end{array}\\ \hline\end{array}$

Sebagai catatan:

$\begin{array}{rl}& \left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc\\ & \text{dan}\\ & \left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|\end{array}$

$\boxed{\text{CONTOH SOAL}}$

Mari kita buka lagi contoh sebelumnya dengan soal yang sama di SINI

dan kearang penyelesaian dari soal tersebut akan diselesaikan dengan cara determinan matriks (cara Cramer sesuai nama penemunya) berikut:

$\begin{array}{l}\\ 1.& \text{Tentukanlah dengan metode matriks}\\ & \text{(cara Cramer) SPLDV berikut}:\\ & \{\begin{array}{ll}2x-y& =7\\ x-y& =-1\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}x& ={\displaystyle \frac{\left|\begin{array}{cc}7& -1\\ -1& -1\end{array}\right|}{\left|\begin{array}{cc}2& -1\\ 1& -1\end{array}\right|}=\frac{7(-1)-(-1).(-1)}{2.(-1)-(-1).1}}\\ & ={\displaystyle \frac{-7-1}{-2+1}=\frac{-8}{-1}=8}\\ y& ={\displaystyle \frac{\left|\begin{array}{cc}2& 7\\ 1& -1\end{array}\right|}{\left|\begin{array}{cc}2& -1\\ 1& -1\end{array}\right|}=\frac{2(-1)-(7).1}{2.(-1)-(-1).1}}\\ & ={\displaystyle \frac{-2-7}{-2+1}=\frac{-9}{-1}=9}\\ \text{J}& \text{adi}(x,y)=(8,9)\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Tentukanlah dengan metode matriks}\\ & \text{(cara Cramer) SPLTV berikut}:\\ & \{\begin{array}{ll}2x-y+z& =-4\\ 2x-y-2z& =-3\\ x+3y-z& =0\end{array}\\ \\ & \text{Jawab}:\\ & \begin{array}{rl}x& ={\displaystyle \frac{\left|\begin{array}{ccc}-4& -1& 1\\ -3& -1& -2\\ 0& 3& -1\end{array}\right|}{\left|\begin{array}{ccc}2& -1& 1\\ 2& -1& -2\\ 1& 3& -1\end{array}\right|}}\\ x& ={\displaystyle \frac{-4\left|\begin{array}{cc}-1& -2\\ 3& -1\end{array}\right|+1\left|\begin{array}{cc}-3& -2\\ 0& -1\end{array}\right|+1\left|\begin{array}{cc}-3& -1\\ 0& 3\end{array}\right|}{2\left|\begin{array}{cc}-1& -2\\ 3& -1\end{array}\right|+1\left|\begin{array}{cc}2& -2\\ 1& -1\end{array}\right|+1\left|\begin{array}{cc}2& -1\\ 1& 3\end{array}\right|}}\\ & ={\displaystyle \frac{-4(1+6)+1(3-0)+1(-9-0)}{2(1+6)+1(-2+2)+1(6+1)}}\\ & ={\displaystyle \frac{-28+3-9}{14+0+7}}\\ & =\frac{-34}{21}\\ y& =....\\ z& =....\\ \text{J}& \text{adi}(x,y,y)=(-{\displaystyle \frac{34}{21},\frac{3}{7},-\frac{1}{3}})\end{array}\end{array}$

DAFTAR PUSTAKA

1. Johanes, Kastola & Sulasim. 2006. Kompetensi Matematika 3A SMA Kelas XII Semester Pertama. Jakarta: YUDHISTIRA